Yes that seems logic. If irc the Romans used to greet by grabbing each other's right wrist, according to the author to assure each other they had no daggers up their sleeves. Do togas or tunics have sleeves?

Were it the Romans who started giving hands?
Or is there an Egyptian glyph somewhere depicting two brothers high-fiving on a pyramid?

The specific behaviour that the researchers focused on was the 'grooming handclasp', a behaviour where two chimpanzees clasp onto each other's arms, raise those arms up in the air, and groom each other with their free arm. This behaviour has only been observed in some chimpanzee populations. The question remained whether chimpanzees are instinctively inclined to engage in grooming handclasp behaviour, or whether they learn this behaviour from each other and pass it on to subsequent generations.

Humans in the past used to groom one another and themselves as it was more common for people to have ticks. When small children greet they'll often hug and touch people. Shaking hands seems like a more formal and less intrusive method of making physical contact.

Where does shaking hands as a form of greeting come from?
Who were the first to do it as recorded in history and why?
Do other primates shake hands?

As mentioned, if a hand is extended it shows you have no weapons.

Its also a more socially acceptable way for the men of the time to size each other up from the strength of their grip, how dominating they are in their grip and so on.

As for the toga, no they dont have sleeves but they are specifically designed to be utterly useless. Thats not an insult, a toga has no practical function, it will fall off if your not careful so it proves that your one of the intellectual elite who can afford to wear non-practical clothing because you never need to raise a finger to work.
Generally speaking you need one hand to keep the damn thing controlled and not falling off, the other hand is is free for gesturing in oratory or whatever, if your shaking hands it kind of limits the chance youve got a dagger hidden away because the other hand is probably holding your clothes on.